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I am stuck with the following question,

How many ways can 8 people be seated in a row? if there are 4 men and 4 women and no 2 men or women may sit next to each other.

I did it as follows, As 4 men and 4 women must sit next to each other so we consider each of them as a single unit. Now we have we 4 people(1 men group, 1 women group, 2 men or women) they can be seated in 4! ways.
Now each of the group of men and women can swap places within themselves so we should multiply the answer with 4!*4!
This makes the total 4!*4!*4! =13824 . Please help me out with the answer. Are the steps clear and is the answer and the method right?

Thanks

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  • $\begingroup$ I don't understand how you get a third factor of $4!$. There are only two ways do divide the eight aligned seats into "man-seats" and "woman-seats" without violating the rule that no two men and no two women amy sit next to each other. $\endgroup$ – Rasmus Oct 23 '11 at 11:49
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If there is a man on the first seat, there has to be a woman on the second, a man on the third so forth. Alternatively, we could start with a woman, then put a man, then a woman and so forth. In any case, if we decide which gender to put on the first seat, the genders for the others seats are forced upon us. So there are only two ways in which we can divide the eight aligned seats into "man-seats" and "woman-seats" without violating the rule that no two men and no two women may sit next to each other.

Once you have chosen where to seat the men and where the women you can permute the two groups arbitrarily, giving $4!$ possibilities each. So in total the number of possible constellations is $$ 2\cdot 4!\cdot 4!=1152. $$

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    $\begingroup$ I was halfway through writing this very answer, you bastard :P $\endgroup$ – Arthur Oct 23 '11 at 11:47
  • $\begingroup$ @Arthur: Sorry to hear that. ;) $\endgroup$ – Rasmus Oct 23 '11 at 11:56
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Look , we have 8 people , 4 of them are men , 4 of them are women , we need to put every man and woman together , it means we will have 4 groups of men and women sitting together. [Group1,Group2,Group3,Group4] , to get Group1 we have 4 possibilities of men , and 4 possibilities of women , then to get Group2 we left with 3 men and 3 women , then in the same manner we got : 4*4*3*3*2*2*1*1 =576 (4!*4!) , so finally we left only with combinations of the groups in the array [Group1,Group2,Group3,Group4], as I got there 4! # of combinations this groups . So I guess this is the answer : 4!*4!*4!

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Is it that no two men and no two women should be seated together. Yep? Then do this.. Permutation of men in 4 places be 4!. This leaves us with 5 gaps(3 in between and 2 on either side) in which we got to arrange 4 women.. P(5,4) i.e. 5! ways. Finally 5!*4! is the answer!.

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